How can I create a hollow sphere with regularly spaced holes?
Instead of the shrinkwrap, there's an older but more precise tool to turn verts into a sphere.

Put the 3D-Cursor where the center of your sphere whould be
Select all verts
Hit Space, type 'to sphere' or press AltShiftS, select that tool and hit 1, Return to make it completely spherical. Most subsurfed spheres aren't really precise, so this tool changes that.

Here's my approach to this:

Making the cookie cutter

Create a mesh that has faces where you want the holes to be (I).
Parent a Cylinder to the Sphere and enable Duplifaces for the sphere (II)
Scale the Cylinder to the desired width
Select the sphere and hit CtrlShiftA to make the duplis real.
Select all Duplis and Shift-Select the middle one, then hit CtrlJ to join them into one single mesh

Making the sphere

Make a sphere to your liking. I used an Icosphere with applied subsurf modifier and 'To Sphere. While in Editmode I gave it depth by duplicating it, scling it in and flipping the normals, so the boolean modifier will be happy.
Put the Cylinder object right where your sphere is and scale it so the cylinders intersect with both inner and outer surface of the sphere. Save your file.
Give the sphere a boolean modifier, set it to Difference and put the cylinder mesh in the Object field to perform the task.
Hit Apply.
Delete, hide or move the Cylinder mesh
Give your sphere an Edge Split modifier and set the shading to smooth.

The advantage is that it works with irregular shapes too.
Try this to get close to the example image
(Bevelling like Mark did is much faster. Do it like he suggested and skip to the bucky ball.)
Start with a level one icosphere

Subdivide it twice

Select one corner vert and use Select (or ShiftG) &gt;&gt; Select Similar &gt;&gt; Amount of Adjacent Faces

Dissolve the verts with X &gt;&gt; Dissolve Vertices.

Enter Face Select Mode.
Select one pentagon and this time use ShiftG&gt; Select Similar &gt;&gt; Polygon Sides.
Invert the selection with CtrlI

Get rid of the excess faces with X &gt;&gt; Limited Dissolve

Select everything and hit the Smooth vertex button a few times. (This is the point to which you could jump if you followed Marks Tutorial. My approach is pre-vertex beveling)
In vertex select mode, subdivide two times. This extra topology is useful to get a round sphere.

Inset individual faces (II)
Delete the big faces with the help of ShiftG&gt; select similar.

Subdivide again. Select everything and use AltShiftS&gt; To Sphere with factor of 1.0

Extrude it four times and scale inwards each time.

Give it a subsurf modifier and enjoy the result.